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Drawing Pfaffian Graphs Drawing Pfaffian Graphs Serguei Norine School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Abstract. We prove that a graph is Pfaffian if and only if it can be drawn in the plane (possibly with crossings) so that every perfect match- ing intersects itself an even number of times. 1 Introduction Inthispaperweproveatheoremthatconnects Pfaffian orientations with the parity of the numbers of crossings in planar drawings. The proof is elementary, but it has other consequences and raises interesting questions. Before we can state the theorem we need some definitions. All graphs considered in this paper are finite and have no loops or multiple edges. For a graph G we denote its edge set by E(G). A labeled graph is a graph with vertex-set {1, 2,...,n} for some n.Ifu and v are vertices in a graph G,then uv denotes the edge joining u and v and directed from u to v if G is directed. A perfect matching is a set of edges in a graph that covers all vertices exactly once. Let G be a directed labeled graph and let M = {u1v1,u2v2,...,ukvk} be a perfect matching of G. Define the sign of M to be the sign of the permutation 1234...2k − 12k . u1 v1 u2 v2 ... uk vk Note that the sign of a perfect matching is well-defined as it does not depend on the order in which the edges are written. We say that a labeled graph G is Pfaffian if there exists an orientation D of G such that the signs of all perfect matchings in D are positive, in which case we say that D is a Pfaffian orientation of G. An unlabeled graph G is Pfaffian if it is isomorphic to a labeled Pfaffian graph. It is well-known and also follows from Theorem 1 below that in that case every labeling of G is Pfaffian. The importance of Pfaffian graphs will be discussed in the next section. By a drawing Γ of a graph G we mean an immersion of G in the plane such that edges are represented by homeomorphic images of [0, 1] not containing vertices in their interiors. Edges are permitted to intersect, but there are only finitely many intersections and each intersection is a crossing. For edges e, f of adrawingΓ let cr(e, f) denote the number of times the edges e and f cross. For a perfect matching M let crΓ (M), or cr(M) if the drawing is understood from context, denote cr(e, f), where the sum is taken over all unordered pairs of distinct edges e, f ∈ M. J. Pach (Ed.): GD 2004, LNCS 3383, pp. 371–376, 2004. c Springer-Verlag Berlin Heidelberg 2004 372 Serguei Norine The following theorem is the main result of this paper. The proof will be presented in Section 3. Theorem 1. A graph G is Pfaffian if and only if there exists a drawing of G in the plane such that cr(M) is even for every perfect matching M of G. The “if” part of this theorem as well as the “if” part of its generalization (Theorem 3) was known to Kasteleyn [4] and was proved by Tesler [14]; however our proof of this part is different. The “only if” part is new. 2 Pfaffian Graphs Pfaffian orientations have been introduced by Kasteleyn [2–4], who demonstrated that one can enumerate perfect matchings in a Pfaffian graph in polynomial time. We say that an n × n matrix A(D)=(aij )isaskew adjacency matrix of a directed labeled graph D with n vertices if 1ifij ∈ E(D), − ∈ aij = 1ifji E(D), 0otherwise. Let A be a skew-symmetric 2n × 2n matrix. For each partition P = {{i1,j1}, {i2,j2},...,{in,jn}} of the set {1, 2,...,2n} into pairs, define 12...2n − 12n aP = sgn ai1j1 ...ainjn . i1 j1 ... in jn Note that aP is well defined as it does not depend on the order of the pairs in the partitions nor on the order in which the pairs are listed. The Pfaffian of the matrix A is defined by Pf(A)= aP , P where the sum is taken over all partitions P of the set {1, 2,...,2n} into pairs. Note that if D is a Pfaffian orientation of a labeled graph G then Pf(A(D)) is equal to the number of perfect matchings in G. One can evaluate the Pfaffian efficiently using the following identity from linear algebra: for a skew-symmetric matrix A det(A)=(Pf(A))2. Thus the number of perfect matchings, and more generally the generating func- tion of perfect matchings of a Pfaffian graph, can be computed in polynomial time. The problem of recognizing Pfaffian bipartite graphs is equivalent to many problems of interest outside graph theory, eg. the P´olya permanent problem [11], Drawing Pfaffian Graphs 373 the even circuit problem for directed graphs [15], or the problem of determin- ing which real square matrices are sign non-singular [5], where the latter has applications in economics [13]. The complete bipartite graph K3,3 is not Pfaffian. Each edge of K3,3 belongs to exactly two perfect matchings and therefore changing an orientation of any edge does not change the parity of the number of perfect matchings with negative sign. One can easily verify that for some (and therefore for every) orientation of K3,3 this number is odd. In fact, Little [6] proved that a bipartite graph is Pfaffian if and only if it does not contain an “even subdivision” H of K3,3 such that G \ V (H)hasa perfect matching. A structural characterization of Pfaffian bipartite graphs was given by Robert- son, Seymour and Thomas [12] and independently by McCuaig [7]. They proved that a bipartite graph is Pfaffian if and only if it can be obtained from planar graphs and one specific non-planar graph (the Heawood graph) by repeated ap- plication of certain composition operations. This structural theorem implies a polynomial time algoritheorem for recognition of Pfaffian bipartite graphs. No satisfactory characterization is known for general Pfaffian graphs. The result of this paper was obtained while attempting to find such a description. 3MainTheorem Let Γ be a drawing of a graph G in the plane. We say that S ⊆ E(G)isamarking of Γ if cr(M)and|M ∩ S| have the same parity for every perfect matching M of G. Theorem 1 follows from the following more general result. Theorem 2. For a graph G the following are equivalent: (a) G is Pfaffian; (b) some drawing of G in the plane has a marking; (c) every drawing of G in the plane has a marking; (d) there exists a drawing of G in the plane such that cr(M) is even for every perfect matching M of G. We say that Γ is a standard drawing of a labeled graph G if the vertices of Γ are arranged on a circle in order and every edge of Γ is drawn as a straight line. The equivalence of conditions (a), (b) and (c) of Theorem 2 immediately follows from the next two lemmas. Lemma 1. Let Γ be a standard drawing of a labeled graph G.ThenG is Pfaffian if and only if Γ has a marking. Proof. Let D be an orientation of G.LetM = {u1v1,u2v2,...,ukvk} be a perfect matching of D. The sign of M is the sign of the permutation 1234...2k − 12k P = . u1 v1 u2 v2 ... uk vk 374 Serguei Norine Let i(P ) denote the number of inversions in P ,then sgn(M)=sgn(P )=(−1)i(P ) = sgn(P (j) − P (i)) = ≤ ≤ 1 i<j 2k = sgn((uj − ui)(vj − ui)(uj − vi)(vj − vi)) × ≤ ≤ 1 i<j k × sgn(vi − ui). (1) 1≤i≤k In Γ edges uivi and ujvj cross if and only if each of the two arcs of the circle containing the vertices of Γ with the ends ui and vi contains one of the vertices uj and vj, in other words if and only if sgn((uj − ui)(vj − ui)(uj − vi)(vj − vi)) = −1. Define SD = {uv ∈ E(D)|u>v}. Note that for every S ⊆ E(G)thereexists (unique) orientation D such that S = SD. From (1) we deduce that sgn(M)=(−1)cr(M) × (−1)|M∩S|. Therefore M has a positive sign if and only if cr(M)and|M ∩ S| have the same parity. It follows that D is a Pfaffian orientation of G if and only if SD is a marking of the standard drawing of G. Notice that we have in fact shown that there exists a one-to-one correspon- dence between Pfaffian orientations of a labeled graph and markings of its stan- dard drawing. Lemma 2. Let Γ1 and Γ2 be two drawings of a labeled graph G in the plane. Then Γ1 has a marking if and only if Γ2 has one. Proof. For any n and any two sequences (a1,a2, .., an)and(b1,b2, .., bn)ofpair- wise distinct points in the plane, there clearly exists a homeomorphic transfor- mation of the plane that takes ai to bi for all 1 ≤ i ≤ n. Therefore without loss of generality we assume that the vertices of G are represented by the same points in the plane in both Γ1 and Γ2. It suffices to prove the statement of the lemma for drawings Γ1 and Γ2 that differ only in the position of a single edge e = uv.Lete1 and e2 denote the images of e in Γ1 and Γ2 correspondingly.
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